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Math Help - Proving...

  1. #1
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    Proving...

    The question asks:

    Prove that if (v1, ... , vn) is a basis of V, then so is(v1, v2-v1, ... , vn - vn-1).

    I made the initial assumption that if V remained the same number that by subtracting v1, youd always end up with v1, and so on. But obviously I came to the conclusion that this would have to be wrong, because the set of (v1, .. , vn) could represent any numbers. So I have no idea how to tackle this one, any help would be greatly appreciated. Thanks.
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  2. #2
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    If a_i are such that \sum_{i=1} ^{n} \ a_iv_i =0 then a_i =0 for all i. Now suppose there are scalars b_i such that \sum_{i=1} ^{n} \ b_i(v_i-v_{i-1})=0 where v_0=0 then we get \sum_{i=1} ^{n} \ (b_i - b_{i+1})v_i=0 where b_{n+1}=0 then b_n=0, and so b_{n-1}=0 ... b_1=0. Which means they're l.i. and therefore a basis.
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  3. #3
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    Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?
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  4. #4
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    Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
    (This is a std theorm and can be found in most of the texts on linear algebra)
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  5. #5
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    Quote Originally Posted by aman_cc View Post
    Any set of 'n' independent vectors which belong to a n-dimensional vector space,V, will be a basis of V.
    (This is a std theorm and can be found in most of the texts on linear algebra)
    Yes I understand that it is a standard theorem, just as we can say X²=4 when x=2. But that is not what I am asking, I am asking how to PROVE this theorem. That is a completely different method. However I appreciate your input.
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  6. #6
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    Quote Originally Posted by GreenDay14 View Post
    Thank you very much Jose, but I am unsure as to whether I can approach the question like this. It would almost appear as if that is proving that they are a basis rather than specifically a basis for V. Would it be possible to get a second opinion on this?
    What is your definition of basis?
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  7. #7
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    It would almost appear as if that is proving that they are a basis rather than specifically a basis for V.
    You are given that the first n vectors form a basis for V so you know that V has dimension n. You are given another n vectors in V. To show that they are a basis for V you need only show that they span V or that they are independent.
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